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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

An estimate for weighted Hilbert transform via square functions


Authors: S. Petermichl and S. Pott
Journal: Trans. Amer. Math. Soc. 354 (2002), 1699-1703
MSC (1991): Primary 42A50; Secondary 42A61
Published electronically: October 26, 2001
MathSciNet review: 1873024
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that the norm of the Hilbert transform as an operator on the weighted space $L^2(w)$ is bounded by a constant multiple of the $3/2$ power of the $A_2$ constant of $w$, in other words by $c\, \sup_I (\langle \omega \rangle_I \langle \omega^{-1} \rangle_I)^{3/2}$. We also give a short proof for sharp upper and lower bounds for the dyadic square function.


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Additional Information

S. Petermichl
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
Address at time of publication: Institute of Advanced Studies, Princeton, New Jersey 08540
Email: stefanie@math.msu.edu

S. Pott
Affiliation: Department of Mathematics, University of York, York YO10 5DD, UK
Email: sp23@york.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02938-5
PII: S 0002-9947(01)02938-5
Keywords: Weighted norm inequalities, square function, Hilbert transform
Received by editor(s): August 15, 2001
Published electronically: October 26, 2001
Additional Notes: The second author gratefully acknowledges support by EPSRC and thanks the Mathematics Department at MSU for its hospitality
Article copyright: © Copyright 2001 American Mathematical Society